Question

the function f is defined for all real values of x. for a constant a, the average rate of change of f from x = a to x = a + 1 is given by the expression 2a + 1. which of the following statements is true? a the average rate of change of f over consecutive equal - length input - value intervals is positive, so the graph of f could be a line with a positive slope. b the average rate of change of f over consecutive equal - length input - value intervals is positive, so the graph of f could be a parabola that opens up. c the average rate of change of f over consecutive equal - length input - value intervals is increasing at a constant rate, so the graph of f could be a line with a positive slope. d the average rate of change of f over consecutive equal - length input - value intervals is increasing at a constant rate, so the graph of f could be a parabola that opens up.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = x_1$ to $x=x_2$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, from $x = a$ to $x=a + 1$, the average rate of change is $2a+1$.

Step2: Analyze the behavior of $2a + 1$

As $a$ increases (since $a\in R$), the value of $2a + 1$ increases. For a line $y=mx + b$, the average rate of change (slope) is constant. For a parabola $y=Ax^{2}+Bx + C$, the average rate of change over intervals of length $1$ is not constant. If the average rate of change of a function over consecutive equal - length input - value intervals is increasing at a constant rate, the function could be a parabola that opens up. The average rate of change of a parabola $y = Ax^{2}+Bx + C$ from $x=a$ to $x=a + 1$ is $A((a + 1)^{2}-a^{2})+B((a + 1)-a)=A(2a + 1)+B$. When $A>0$, the average rate of change is an increasing linear function of $a$.

Step3: Evaluate each option

Option A: For a line, the average rate of change is constant. Since $2a + 1$ is not constant (it depends on $a$), option A is incorrect.
Option B: Just because the average rate of change is positive does not necessarily mean it is a parabola that opens up. We need to consider the non - constancy of the average rate of change.
Option C: A line has a constant average rate of change. Since $2a+1$ is not constant, option C is incorrect.
Option D: Since the average rate of change $2a + 1$ is an increasing linear function of $a$, the function $f(x)$ could be a parabola that opens up.

Answer:

D. The average rate of change of $f$ over consecutive equal - length input - value intervals is increasing at a constant rate, so the graph of $f$ could be a parabola that opens up.